Calculating the input-output dynamic characteristics. Analyzing dynamic systems and designing controllers.

Transcription

1 CHAPTER : REVIEW OF FREQUENCY DOMAIN ANALYSIS The long-term response of a proess is nown as the frequeny response whih is obtained from the response of a omplex-domain transfer funtion. The frequeny response analysis is useful for: Calulating the input-output dynami harateristis. Analyzing dynami systems and designing ontrollers.. Shortut method for Frequeny response Step : For a given s-domain transfer funtion s, set s to get Step : Express in terms of R I using omplex onugate multipliation. Step 3: ompute the amplitude ratio as as p φ tan I / R AR R I and the phase angle. properties of Frequeny response For a omplex transfer funtion suh as: a s b s s L s s s L 3 s The amplitude ratio is simply: a b L 3 L The phase angle is simply: p φ φ φ L φ φ φ a b Bode Plot Bode diagram is used to display where AR and φ are plotted as a funtion of frequeny. These plots are useful for: Rapid analysis of the response harateristi Analyzing stability of losed-loop systems.

2 .4 Examples.4. First Order System Let the proess model be: s s p 4 p 5 I R p 6 Therefore; R 7 I 8 Consequently: AR 9 tan tan φ p Seth and Analysis of the frequeny response: At low frequeny, i.e. <</: AR φ At high frequeny, i.e. >> /:

3 AR / ο φ 9 At the brea point orner frequeny, i.e. /: AR.77; for φ tan 45 ο For qui seth of the frequeny response ompute the slope of the amplitude ratio at high frequeny as: log AR log log log Then plot the frequeny response on a log-log sale as follows: -3 AR/K. φ -6 slope Figure : Frequeny response for a first-order system.4. Seond Order System A general seond order system is given by: p s s ζs Following the above tehnique we an show that: AR ζ ζ φ tan 3

4 For over-damped system, ζ > At low frequeny: AR N AR / K At high frequeny: φ tan AR N AR / K φ tan 8 o At orner frequeny: AR N AR / K.77 φ tan 9 o The slope at high frequeny: log AR log log log The seth is almost similar to that of a first order system: AR/K.. slope φ Figure : Frequeny response for seond-order system For under-damped system, the bode plot is almost the same exept that it may have a maximum amplitude ratio. Taing the derivative of equation with respet to, gives: d d AR ζ [ ζ ] 3/ 4

5 Setting the last equation to zero gives: ζ 5 Substituting the value of into equation yields: AR max ζ ζ 6 Thus, the Bode plot loos lie:. AR/K slope φ Figure 3: Frequeny response for under-damped system The frequeny at whih the maximum our equation 5 is nown as the resonant frequeny beause at that frequeny the sinusoidal output response has the maximum amplitude for a given sinusoidal input..4.3 Time delay p s e θs θw p e osθ sin θ 7 8 AR os θ sin θ 9 osθ φ tan θ sin θ Therefore, the seth of this frequeny system is:

6 AR/K -8 φ Figure 4: Frequeny response for a time delay.4.4 Zero Lead proesses p s s p w R I AR p φ tan 3 4 At low frequenies, At intermediate frequeny, / At high frequenies, >> AR φ tan - AR φ tan 45 - o AR φ tan 9 - The slope: log AR log The qui seth of the frequeny response is:

7 9 6 AR/K φ 3. slope Figure 5: Frequeny response of a lead proess Comments Lag proesses: A lag proesses always have a negative phase angle whih indiates that the output follows or lag the input by φ. For a lag proesses, the amplitude ratio always approahes zero for high frequenies. Lead proesses zero proesses The amplitude ratio beomes very large at high frequenies. The phase angle is positive exept in the presene of RHP zeros. Proesses with RHP zeros or time delay are nown as nonminimum phase systems beause they exhibit phase lag..4.5 Composite system.5s 5.5s e s s 4s Aording to the properties in equations and we have: AR φ tan 5.5 e log AR log5 tan tan tan.5.5 log.5 log log log For qui seth the bode plot we have: For the amplitude ratio, simply aumulate the slopes at high frequenies desending from the largest time onstant: This means the slope should be: log AR log log4 log. 5 5

8 Frequeny slope < < / / < < /4 /4 < < /.5 /.5 < < For the phase angle simply sum all phase angles as shown in Figure 6.. AR/K φ Figure 6: Frequeny response for omposite system.5 Bode Stability Criteria A losed-loop system is unstable if the frequeny response of the openloop transfer funtion ol vp m has an amplitude ratio greater than one at the ritial frequeny. Otherwise the lose-loop system is stable. The ritial frequeny is defined as to be the frequeny at whih the open-loop phase angle is -8 o.

9 Limitations: Bode stability riteria an not be used for unstable proess, whih have multiple ritial frequenies. For these types of proesses, Nyquist stability riteria might be used. Example.: iven p s ; 3 s ; v s.`; m s.5s OL s.5s 3 The bode plot for this system at three different values for is shown o in Figure 7. AR - - Frequeny phase angle w Frequeny Figure 7: Bode diagram for three values for.

10 It obvious that: At the system is stable beause AR << at At 4 the system is marginally stable beause AR at At the system is unstable beause AR >> at.6 Effet of ontroller on frequeny response A typial frequeny response for PID ontroller: 3 AR Frequeny 9 PD phase angle -9 PID PI Frequeny Figure 8: Typial frequeny response for three modes of PID ontroller Integral ation is inluded in ontroller to eliminate offset. However, it adds phase lag maing the system less stable. In this ase, the phase angle urve dereases rapidly, thus, the hanes for the phase lag of high-order system with PI ontroller to ross -8 o at low frequenies is higher.

11 Derivative ation adds phase lead improving stability and allowing higher gains to be used to improve the losed-loop response. Tuning is benefiial to ahieve: A large value for is also desirable sine it indiates small lose-loop response time. It only desirable that the amplitude ratio be small at but it an be inreased at other frequenies to improve ontrol system performane. Example. 5 p s ; s ; v s m s s.5s B AR - C A Frequeny phase angle Frequeny Figure 9: Bode Diagram for Example 5.; urve A: proportional ontroller; urve B: PI ontroller with.4, I.; urve C: PI ontroller with.4, I It is lear from urve A in Figure 9, that a proportional ontrol does not add any phase lag. Note that the ritial frequeny does not exist beause φ > -8 o for all frequenies. Hene, an be extremely large and the losed-loop system will always be stable. u is infinity in this ase.

12 The inlusion of integral ation in the ontroller an ause the losed-loop system to beome unstable. Curve B in Figure 9 shows that the phase angle rosses -8 o with AR >> for.4/.s. Curve C indiates that when I is inreased to, a stable losedloop response results for all values of beause there is no ritial frequeny..7 ain and Phase Margin As a measure or relative stability, the term gain and phase margins are used. The gain margin is defined as: M AR 6 Here AR is the value of the open-loop amplitude ratio at the ritial frequeny. Sine AR must be less than one for stability, then M > is a stability requirement. If g is the frequeny at whih the open-loop gain AR is unity and φ g is the phase angle at that frequeny, then: PM 8 φ g 7. AR ain margin φ g 8 Phase margin. g Figure : ain and Phase margins on Bode plot

13 Remars: Controller manufaturers reommend that a well-tuned ontroller has a gain margin of.7 to and a phase margin of 3 to 45 o. M and PM an be used to provide a omparison between good performane and stability. Large values for M and PM ause sluggish lose-loop response, while small values result in a less sluggish, more osillatory response. The onept of gain and phase margin does not apply for proesses with multiple ritial frequenies..8 Closed-loop Frequeny domain The amplitude ratio and phase angle for losed-loop response is given by: M y r v v p p m 8 ψ y r v p p p 9 v p m M p bw Figure : Closed-loop response Comments: M should be unity as for set point and zero as for disturbane indiating no offset. For set point, M should be maintained as unity up to as high frequeny as possible, while for disturbane, M should be minimized over as wide a frequeny range as possible. This is to ensure a rapid approah to steady state.

14 The pea amplitude ratio at the resonant frequeny should no larger than.5 orresponding to a damping ratio of ζ.5 The ontroller should be tuned suh that p is as large as possible. A large value for p implies faster response to set point. The bandwidth bw is the frequeny at whih M /. 77. A large value for bw indiates a relatively fast response with a short rise time. b brea orner frequeny / ritial frequeny 8 tan - p resonane frequeny M max p